The paper focuses on development of the analytical theory to assess spatial distribution of energy released during propagation of the fast electron beam in a gas, in particular in the air at electron energies of 1400 keV. An approach adopted by authors [2, 3] to study inelastic deceleration of electrons in the air is further developed here. As the inelastic interaction in most cases leads to energy relaxation while elastic interaction causes distribution isotropization over directions, the first task solved in the paper is finding the electron distribution function including only elastic collisions. In the final part of this paper an analytical solution to this task is presented with account of both types of electron deceleration in the air.

The calculations show that when elastic collisions are taken into account this leads to increased spatial density of energy release and to narrowing of the primary energy release region of the fast electrons, as compared to calculations accounting for only inelastic deceleration.